Holditch-Type Theorem for Non-Linear Points in Generalized Complex Plane $\mathbb{C}_{p}$

Year 2018, Volume: 1 Issue: 4, 239 - 243, 20.12.2018

Tülay Erişir , Mehmet Ali Güngör

https://doi.org/10.32323/ujma.430853

Cited By: 4

Abstract

The generalized complex number system and generalized complex plane were studied by Yaglom [22], [23] and Harkin [7]. Moreover, Holditch-type theorem for linear points in $\mathbb{C}_{p}$ were given by Eri\c{s}ir et al. [6]. The aim of this paper is to find the answers of the questions ''How is the polar moments of inertia calculated for trajectories drawn by non-linear points in $\mathbb{C}_{p}$?'', ''How is Holditch-type theorem expressed for these points in $\mathbb{C}_{p}$?'' and finally ''Is this paper a new generalization of [6]?''.

Keywords

Generalized complex plane, Holditch-type theorem, The polar moment of inertia

References

• [1] W. Blaschke, H. R. Müller, Ebene kinematik verlag oldenbourg, München, 1956.
• [2] A. Broman, A fresh look at a long-forgotten theorem, Math. Mag., 54(3) (1981), 99–108.
• [3] M. Düldül, Computation of polar moments of inertia with Holditch type theorem, Appl. Math. E-Notes, 8 (2008), 271–278.
• [4] T. Eris¸ir, M. A. Güngör, Cauchy-Length formula and Holditch theorem in the generalized complex plane Cp, Int. Electron. J. Geom., (Accepted).
• [5] T. Erişir, M. A. Güngör, M. Tosun, A new generalization of the Steiner formula and the Holditch theorem, Adv. Appl. Clifford Algebr., 26(1) (2016), 97–113.
• [6] T. Eris¸ir, M. A. Güngör, M. Tosun, The Holditch-type theorem for the polar moment of inertia of the orbit curve in generalized complex plane, Adv. Appl. Clifford Algebr., 26(4) (2016), 1179–1193.
• [7] A. A. Harkin, J. B. Harkin, Geometry of generalized complex numbers, Math. Mag., 77(2) (2004), 118–129.
• [8] N. (Bayrak) Gürses, S. Yüce, One-parameter planar motions in affine Cayley-Klein planes, Eur. J. Pure Appl. Math., 7(3) (2014), 335–342.
• [9] N. (Bayrak) Gürses, S. Yüce, One-parameter planar motions in generalized complex number plane Cj , Adv. Appl. Clifford Algebr., 25(4) (2015), 889–903.
• [10] L. Hering, Satze vom Holditch-Typ f¨ur ebene kurven, Elem. Math., 38 (1983), 39–49.
• [11] H. Holditch, Geometrical theorem, Quat. J. Pure Appl. Math., 2 (1858).
• [12] H. R. Müller, Uber Tragheitsmomente bei Steinerscher Massenbelegung, Abh. Braunschw. Wiss. Ges., 29 (1978), 115–119.
• [13] L. Parapatits, F. E. Schuster, The Steiner formula for Minkowski valuations, Adv. Math., 230 (2012), 978–994.
• [14] H. Potmann, Holditch-Sicheln, Arc. Math., 44 (1985), 373–378.
• [15] H. Potmann, Zum satz von Holditch in der Euklidischen ebene, Elem. Math., 41 (1986), 1–6.
• [16] J. Steiner, Gesammelte werke II, Berlin, 1881.
• [17] S. Yüce, M. Düldül, N. Kuruoğlu, On the generalizations of the polar inertia momentum of the closed curves obtained kinematically, Studia Sci. Math. Hungar., 42(1) (2005), 73–78.
• [18] S. Yüce, N. Kuruoğlu, Cauchy formulas for enveloping curves in the Lorentzian plane and Lorentzian kinematics, Result. Math., 54 (2009), 199–206.
• [19] S. Yüce, N. Kuruoğlu, Holditch-type theorems for the planar Lorentzian Motions, Int. J. Pure Appl. Math., 17(4) (2004), 467–471.
• [20] S. Yüce, N. Kuruoğlu, Holditch-Type theorems under the closed planar homothetic motions, Ital. J. Pure Appl. Math., 21 (2007), 105–108.
• [21] S. Yüce, N. Kuruoğlu, Steiner formula and Holditch-Type theorems for homothetic Lorentzian motions, Iran. J. Sci. Technol. Trans. A Sci., 31 (2007), 207–212.
• [22] I. M. Yaglom, A Simple Non-Euclidean Geometry and its Physical Basis, Springer-Verlag, New-York, 1979.
• [23] I. M. Yaglom, Complex Numbers in Geometry, Academic Press, New York, 1968.